Ah, I really recommend selling the 780 and cancel the 1060 3GB purchase (if you can) if you're a serious gamer (that's different than heavy gamer).
I'm playing Gears of War 4 and the VRAM is passing 3GB by far (total use with the desktop reaches 4GB) causing occasional stutter with new textures showing up so I had to reduce two texture settings from ultra to high to get ~2.4GB game usage to reduce the stutter (still exceeded 3GB a little sometimes with the desktop usage). The memory bus width on my GTX 680 4GB is 256-bit, higher than that on the 1060 3GB which means even slower texture swapping let alone the less size that causes using the much slower system RAM (which reached a crazy 14GB total of texture buffering at some point because of that!).
The 780 has faster memory bus width but is it at least 4GB? Games started to use more VRAM lately even at 1080p like with Gears of War 4.
Later models are better in terms of power and support, of course, but performance is always the target for gamers specially if it is noticeable. Try finding some gameplay of the same game on both cards on Youtube to compare. This could help.
And no, the model number indicates direct rule to do with the resolution. A lower model number could be able to use higher resolution that a higher model number.